DOCTOR OF PHILOSOPHYDepartment of Civil EngineeringCollege of Engineering

KANSAS STATE UNIVERSITYManhattan, Kansas2012

AbstractThe analysis of concrete columns using unconfined concrete models is a well establishedpractice. On the other hand, prediction of the actual ultimate capacity of confined concretecolumns requires specialized nonlinear analysis. Modern codes and standards are introducing theneed to perform extreme event analysis. There has been a number of studies that focused on theanalysis and testing of concentric columns or cylinders. This case has the highest confinementutilization since the entire section is under confined compression. On the other hand, theaugmentation of compressive strength and ductility due to full axial confinement is notapplicable to pure bending and combined bending and axial load cases simply because the areaof effective confined concrete in compression is reduced. The higher eccentricity causes smallerconfined concrete region in compression yielding smaller increase in strength and ductility ofconcrete. Accordingly, the ultimate confined strength is gradually reduced from the fullyconfined value fcc(at zero eccentricity) to the unconfined value fc(at infinite eccentricity) as afunction of the compression area to total area ratio. The higher the eccentricity the smaller theconfined concrete compression zone. This paradigm is used to implement adaptive eccentricmodel utilizing the well known Mander Model and Lam and Teng Model.Generalization of the moment of area approach is utilized based on proportional loading, finitelayer procedure and the secant stiffness approach, in an iterative incremental numerical model toachieve equilibrium points of P-εand M-ϕresponse up to failure. This numerical analysis isadaptod to asses the confining effect in circular cross sectional columns confined with FRP andconventional lateral steel together; concrete filled steel tube (CFST) circular columns andrectangular columns confined with conventional lateral steel. This model is validated againstexperimental data found in literature. The comparison shows good correlation. Finally computer

software is developed based on the non-linear numerical analysis. The software is equipped withan elegant graphics interface that assimilates input data, detail drawings, capacity diagrams anddemand point mapping in a single sheet. Options for preliminary design, section andreinforcement selection are seamlessly integrated as well. The software generates 2D interactiondiagrams for circular columns, 3D failure surface for rectangular columns and allows the user todetermine the 2D interaction diagrams for any angle α between the x-axis and the resultantmoment. Improvements to KDOT Bridge Design Manual using this software with reference toAASHTO LRFD are made. This study is limited to stub columns.

A DISSERTATIONSubmitted in partial fulfillment of the requirements for the degree

DOCTOR OF PHILOSOPHYDepartment of Civil EngineeringCollege of EngineeringKANSAS STATE UNIVERSITYManhattan, Kansas2012Approved by:

Major ProfessorHayder Rasheed

Copyright

AHMED MOHSEN ABD EL FATTAH2012

AbstractThe analysis of concrete columns using unconfined concrete models is a well establishedpractice. On the other hand, prediction of the actual ultimate capacity of confined concretecolumns requires specialized nonlinear analysis. Modern codes and standards are introducing theneed to perform extreme event analysis. There has been a number of studies that focused on theanalysis and testing of concentric columns or cylinders. This case has the highest confinementutilization since the entire section is under confined compression. On the other hand, theaugmentation of compressive strength and ductility due to full axial confinement is notapplicable to pure bending and combined bending and axial load cases simply because the areaof effective confined concrete in compression is reduced. The higher eccentricity causes smallerconfined concrete region in compression yielding smaller increase in strength and ductility ofconcrete. Accordingly, the ultimate confined strength is gradually reduced from the fullyconfined value fcc(at zero eccentricity) to the unconfined value fc(at infinite eccentricity) as afunction of the compression area to total area ratio. The higher the eccentricity the smaller theconfined concrete compression zone. This paradigm is used to implement adaptive eccentricmodel utilizing the well known Mander Model and Lam and Teng Model.Generalization of the moment of area approach is utilized based on proportional loading, finitelayer procedure and the secant stiffness approach, in an iterative incremental numerical model toachieve equilibrium points of P-εand M-ϕresponse up to failure. This numerical analysis isadaptod to asses the confining effect in circular cross sectional columns confined with FRP andconventional lateral steel together, concrete filled steel tube (CFST) circular columns andrectangular columns confined with conventional lateral steel. This model is validated againstexperimental data found in literature. The comparison shows good correlation. Finally computer

software is developed based on the non-linear numerical analysis. The software is equipped withan elegant graphics interface that assimilates input data, detail drawings, capacity diagrams anddemand point mapping in a single sheet. Options for preliminary design, section andreinforcement selection are seamlessly integrated as well. The software generates 2D interactiondiagrams for circular columns, 3D failure surface for rectangular columns and allows the user todetermine the 2D interaction diagrams for any angle α between the x-axis and the resultantmoment. Improvements to KDOT Bridge Design Manual using this software with reference toAASHTO LRFD are made. This study is limited to stub columns

The author expresses his gratitude to his supervisor, Dr. Hayder Rasheed who wasabundantly helpful and offered invaluable assistance, support and guidance. Deepest gratitudeare also due to the members of the supervisory committee, Dr. Asad Esmaeily, Dr Hani Melhem,Dr Sutton Stephens and Dr Brett DePaola,.The author would also like to convey thanks to Kansas Department of transportation forproviding the financial means of this research

The author wishes to express his love and gratitude to his beloved family; for their understanding& endless love, through the duration of his studies. Special thanks to his mother Dr AmanyAboellil for her support

xxviii

DedicationThis work is dedicated to the memory of the ones who couldnt make it.

1

Chapter 1 - Introduction1-1 BackgroundColumns are considered the most critical elements in structures. The unconfined analysisfor columns is well established in the literature. Structural design codes dictate reduction factorsfor safety. It wasnt until very recently that design specifi cations and codes of practice, likeAASHTO LRFD, started realizing the importance of introducing extreme event load cases thatnecessitates accounting for advanced behavioral aspects like confinement. Confinement addsanother dimension to columns analysis as it increases the columns capacity and ductility.Accordingly, confinement needs special non linear analysis to yield accurate predictions.Nevertheless the literature is still lacking specialized analysis tools that take into accountconfinement despite the availability of all kinds of confinement models. In addition the literaturehas focused on axially loaded members with less attention to eccentric loading. Although thelatter is more likely to occur, at least with misalignement tolerances, the eccentricity effect is notconsidered in any confinement model available in the literature.It is widely known that code Specifications involve very detailed design procedures thatneed to be checked for a number of limit states making the task of the designer very tedious.Accordingly, it is important to develop software that guide through the design process andfacilitate the preparation of reliable analysis/design documents.

1-2 ObjectivesThis study is intended to determine the actual capacity of confined reinforced concretecolumns subjected to eccentric loading and to generate the failure envelope at three different2

levels. First, the well-known ultimate capacity analysis of unconfined concrete is developedas a benchmarking step. Secondly, the unconfined ultimate interaction diagram is scaleddown based on the reduction factors of the AASHTO LRFD to the design interactiondiagram. Finally, the actual confined concrete ultimate analysis is developed based on a neweccentricity model accounting for partial confinement effect under eccentric loading. Theanalyses are conducted for three types of columns; circular columns confined with FRP andconventional transverse steel, circular columns confined with steel tubes and rectangularcolumns confined with conventional transverse steel. It is important to note that the presentanalysis procedure will be benchmarked against a wide range of experimental and analyticalstudies to establish its accuracy and reliability.It is also the objective of this study to furnish interactive software with a user-friendlyinterface having analysis and design features that will facilitate the preliminary design ofcircular columns based on the actual demand. The overall objectives behind this research aresummarized in the following points:- Introduce the eccentricity effect in the stress-strain modeling- Implement non-linear analysis for considering the confinement effects on columns actualcapacity- Test the analysis for three types of columns; circular columns confined with FRP andconventional transverse steel, circular columns confined with steel tubes and rectangularcolumns confined with conventional transverse steel.- Generate computer software that helps in designing and analyzing confined concretecolumns through creating three levels of Moment-Force envelopes; unconfined curve,design curve based on AASHTO-LRFD and confined curve.3

Chapter 2 - Literature ReviewThis chapter reviews four different topics; lateral steel confinement models,Circular Concrete Columns Filled Steel Tubes (CFST) and Rectangular Columnssubjected to biaxial bending and Axial Compression.2-1 Steel Confinement ModelsA comprehensive review of confined models for concrete columns under concentric axialcompression that are available in the literature is conducted. The models reviewed arechronologically presented then compared by a set of criteria that assess consideration of differentfactors in developing the models such as effectively confined area, yielding strength andductility.2-1-1 Chronological Review of ModelsThe confinement models available are presented chronologically regardless of theircomparative importance first. After that, discussion and categorization of the models are carriedout and conclusions are made. Common notation is used for all the equations for the sake ofconsistency and comparison.2-1-1-1 NotationAs:the cross sectional area of longitudinal steel reinforcementAst:the cross sectional area of transverse steel reinforcementAe:the area of effectively confined concrete5

Acc:the area of core within centerlines of perimeter spirals or hoo ps excluding area oflongitudinal steelb: the confined width (core) of the sectionh: the confined height (core) of the sectionc: center-to-center distance between longitudinal barsds: the diameter of longitudinal reinforcementdst: the diameter of transverse reinforcementD: the diameter of the columndsthe core diameter of the columnfcc: the maximum confined strengthfc:the peak unconfined strengthfl:the

ρs: the volumetric ratio of lateral steel to concrete coreρl:the ratio of longitudinal steel to the gross sectional areaρ: the volumetric ratio of lateral + longitudinal steel to concrete core

Richart, Brandtzaeg and Brown (1929)

Richart et als. (1929) model was the first to capture the proportional relationshipbetween the lateral confined pressure and the ultimate compres sive strength of confinedconcrete.lcccfkff1'+=2-1The average value for the coefficient k1,which was derived from a series of short columnspecimen tests, came out to be (4.1). The strain corresponding to the peak strengthεcc(seeMander et al. 1988) is obtained using the following function:+='21clcoccffkεε

125kk=2-2whereεcois the strain corresponding to

fc,k2is the strain coefficient of the effective lateralconfinement pressure. No stress-strain curve graph was proposed by Richart et al (1929).

Chan (1955)

A tri-linear curve describing the stress-strain relationship was suggested by Chan (1955)based on experimental work. The ratio of the volume of steel ties to concrete core volume andconcrete strength were the only variables in the experimental work done. Chan assumed that OAapproximates the elastic stage and ABC approximates the plastic stage, Figure (2-1). Thepositions of A, B and C may vary with different concrete variables. Chan assumed three different7

Blume et al. (1961) were the first to impose the effect of the yield strength for thetransverse steel fyhin different equations defining the model. The model generated, Figure (2-2),has ascending straight line with steep slope starting from the origin till the plain concrete peakstrength fcand the corresponding strain εco, then a less slope straight line connect the latter pointand the confined concrete peak strength fccand εcc. Then the curve flatten till εcushfAffyhstccc1.485.0'+= for rectangular columns 2-3psipsifcco6'1040022.0 +=ε2-4yccεε5=2-5sucuεε5=2-6

Figure 2-1: General Stress-Strain curve by Chan (1955)λ2Ec

λ1Ec

StrainStressufpfefOABCeεpεuεγ1Ec

γ2Ec

8

StressStrain0.85f'cfccεcoεccεcu

Figure 2-2: General Stress-Strain curve by Blume et al. (1961)

where εyis the strain at yielding for the transverse reinforcement, Astis the cross sectional area oftransverse steel reinforcement ,h is the confined cross sectional height,εsuis the strain oftransverse spiral reinforcement at maximum stress andεcuis the ultimate concrete strain.

Roy and Sozen (1965)

Based on their experimental results, which were controlled by two variables; ties spacingand amount of longitudinal reinforcement, Roy and Sozen (1965) concluded that there is noenhancement in the concrete capacity by using rectilinear ties. On the other hand there wassignificant increase in ductility. They proposed a bilinear ascending-descending stress straincurve that has a peak of the maximum strength of plain concrete fcand corresponding strain εco

with a value of 0.002. The second line goes through the point defined byε50till it intersects withthe strain axis. The strain ε50was suggested to be a function of the volumetric ratio of ties toconcrete core ρs, tie spacing s and the shorter side dimension b (see Sheikh 1982).9

sbs4'350ρε= 2-7

Soliman and Yu (1967)

Soliman and Yu (1967) proposed another model that emerged from experimental results.The main parameters involved in the work done were tie spacing s, a new term represents theeffectiveness of ties so, the area of ties Ast, and finally section geometry, which has three differentvariables; Accthe area of confined concrete under compression, Acthe area of concrete undercompression and b. The model has three different portions as shown in Figure (2-3). Theascending portion which is represented by a curve till the peak point (fc, εce). The flat straight-line portion with its length varying depending on the degree of confinement. The last portion is adescending straight line passing through (0.8 fc, εcf) then extending down till an ultimate strain.()20028.045.04.1BssAssAAAqstostccc+−−=2-8()qffccc05.019.0'+= 2-9610*55.0−=cccefε2-10

)1(0025.0 qcs+=ε2-11)85.01(0045.0 qcf+=ε2-12

where q refers to the effectiveness of the transverse reinforcement , sois the vertical spacing atwhich transverse reinforcement is not effective in concrete confinement and B is the greater of band 0.7 h.10

StressStrain'cf'8.0cfceεcsεcfε

Sargin (1971)

Sargin conducted experimental work on low and medium strength concrete with no longitudinalreinforcement. The transverse steel that was used had different size and different yield andultimate strength. The main variables affecting the results were the volumetric ratio of lateralreinforcement to concrete core ρs, the strength of plain concrete fc, the ratio of tie spacing to thewidth of the concrete core and the yield strength of the transverse steel fyh.()+−+−+=22'3)2(11mxxAxmAxfkfcc2-13where m is a constant controlling the slope of the descending branch:'05.08.0cfm −= 2-14cccxεε=2-15'3 ccccfkEAε=2-16'3245.010146.01cyhsffbskρ−+=2-17Figure 2-3: General Stress-Strain curve by Soliman and Yu (1967)11

As Roy and Sozen (1965) did, Kent and Park (1971) assumed that the maximum strengthfor confined and plain concrete is the same fc. The suggested curve, Figure (2-4), starts from theorigin then increases parabolically (Hognestads Parabola) till the peak at fcand thecorresponding strainεcoat 0.002. Then it descends with one of two different straight lines. Forthe confined concrete, which is more ductile, it descends till the point (0.5 fc, ε50c) and continuesdescending to 0.2fc followed by a flat plateau. For the plain concrete it descends till the point(0.5 fc, ε50u) and continue descending to 0.2fcas well without a flat plateau. Kent and Parkassumed that confined concrete could sustain strain to infinity at a constant stress of 0.2 fc

The four constant A, B, C, D were evaluated for the ascending part independently of thedescending one. The four conditions used to evaluate the constants for the ascending part weredY/dX = E0.45/Esecat X=0 Esec= fcc/εcc

where fiandεiare the stress and strain at the inflection point, f2iandε2irefer to a point suchthatcciiiεεεε−=−2and E0.45represents the secant modulus of elasticity at 0.45 fccY = f2i/fccfor X =ε2i/εcc

Muguruma , Watanabe , Katsuta and Tanaka (1980)

StrainStressccfccf45.0ccεiεi2εifif216

Muguruma et al. (1980) obtained their stress-strain model based on experimental workconducted by the model authors, Figure (2-7). The stress-strain model is defined by three zones;Zone 1 from 0-A:22'ccocoiccicEfEfεεεε−+=(kgf/cm2)cocεε≤≤02-40

Zone 2 from A-D()( )( )ccccccoccccccffff −−−+='22εεεε(kgf/cm2)ccccoεεε≤<2-41Zone 3 from D-E( )ccccccuccucccffffεεεε−−−+=(kgf/cm2)cucccεεε≤<2-42()cuccccccufSfεεε+−=2(kgf/cm2) 2-43()2000/100413.0'cuf−=ε(kgf/cm2) 2-44−=WsffCccyhs5.01'ρ2-45whereS is the area surrounded by the idealized stress-strain curve up to the peak stress and W isthe minimum side length or diameter of confined concreteFor circular columns confined with circular hoops:17

Scott et al. (1982) examined specimens by loading at high strain rate to correlate with theseismic loading. They presented the results including the effect of eccentric loading, strainrate, amount and distribution of longitudinal steel and amount and distribution of transversesteel. For low strain rate Kent and Park equations were modified to fit the experimental data−=2'002.0002.02kkkffccccεε

where b is the width of concrete core measured to outside of the hoops. For the high strainrate, the k and Zmwere adapted to)1(25.1'cyhsffkρ+=2-56ksbffZsccm002.043100014529.03625.0''−+−+=ρfcis in MPa 2-57and the maximum strain was suggested to be:19

+=3009.0004.0yhscufρε2-58It was concluded that increasing the spacing while maintaining the same ratio of lateralreinforcement by increasing the diameter of spirals, reduce the efficiency of concreteconfinement. In addition, increasing the number of longitudinal bars will improve the concreteconfinement due to decreasing the spacing between the longitudinal bars.

Sheikh and Uzumeri (1982)

Sheikh and Uzumeri (1982) introduced the effectively confined area as a new term indetermining the maximum confined strength (Soliman and Yu (1967) had trial in effective areaintroduction). In addition to that they, in their experimental work, utilized the volumetric ratio oflateral steel to concrete core, longitudinal steel distribution, strength of plain concrete, and tiesstrength, configuration and spacing. The stress-strain curve, Figure (2-8), was presentedparabolically up to (fcc, εcc), then it flattens horizontally till εcs,and finally it drops linearlypassing by (0.85fcc, ε85) till 0.3 fcc, In that sense, it is conceptually similar to the earlier model ofSoliman and Yu (1967).fccand εcccan be determined from the following equations:cpsccfkf ='cpcpfkf =85.0=pk 2-59'2222215.5173.21stsoccsfbsbncPbkρ−−+=2-606'10*55.0−=csccfkε2-61

where b is the confined width of the cross section, fstis the stress in the lateral confining bar, c iscenter-to-center distance between longitudinal bars,εs85is the value of strain corresponding to85% of the maximum stress on the unloading branch, n is the number of laterally supportedlongitudinal bars, Z is the slope for the unloading part, fcpis the equivalent strength ofunconfined concrete in the column, and Pocc= Kpf'c(Acc- As)

Ahmad and Shah (1982) developed a model based on the properties of hoopreinforcement and the constitutive relationship of plain concrete. Normal weight concrete andlightweight concrete were used in tests that were conducted with one rate of loading. Nolongitudinal reinforcement was provided and the main two parameters varied were spacing andyield strength of transverse reinforcement. Ahmed and Shah observed that the spirals becomeineffective when the spacing exceeds 1.25 the diameter of the confined concrete column. Theyconcluded also that the effectiveness of the spiral is inversely proportional with compressivestrength of unconfined concrete.Ahmad and Shah adapted Sargin model counting on the octahedral failure theory, thethree stress invariants and the experimental results:22)2(1)1(XDXAXDXAYiiii+−+−+=2-65pcnpcsffY =2-66ipiXεε=2-67where fpcsis the most principal compressive stress, fpcnis the most principal compressive strength,εiis the strain in the i-th principal direction andεipis the strain at the peak in the i-th direction.

ipiiEEA =

ippcnipfEε=

Eiis the initial slope of the stress strain curve, Diis a parameter that governs the descendingbranch. When the axial compression is considered to be the main loading, which is typically thecase in concentric confined concrete columns, Equations (2-65), (2-66) and (2-67) become:22

22)2(1)1(DXXAXDAXY+−+−+= 2-68cccffY =2-69cccXεε=2-70secEEAc=2-71

Park, Priestly and Gill (1982)

Park et al. (1982) modified Kent and Park (1971) equations to account for the strengthimprovement due to confinement based on experimental work conducted for four square fullscaled columns (21.7 in2(14 000 mm2) cross sectional area and 10.8 ft (3292 mm) high), Figure(2-9). The proposed equations are as follow:−=2'002.0002.02kkkff